# American Institute of Mathematical Sciences

July & August  2021, 20(7&8): 2857-2883. doi: 10.3934/cpaa.2020245

## The two-component Novikov-type systems with peaked solutions and $H^1$-conservation law

 School of Mathematics and Statistics and Center for Nonlinear Studies, Ningbo University, Ningbo 315211, China

* Corresponding author

Received  June 2020 Revised  July 2020 Published  July & August 2021 Early access  September 2020

Fund Project: This work is supported by the NSF-China grant-11631007 and grant-11971251

In this paper, we provide a classification to the general two-component Novikov-type systems with cubic nonlinearities which admit multi-peaked solutions and $H^1$-conservation law. Local well-posedness and wave breaking of solutions to the Cauchy problem of a resulting system from the classification are studied. First, we carry out the classification of the general two-component Novikov-type system based on the existence of two peaked solutions and $H^1$-conservation law. The resulting systems contain the two-component integrable Novikov-type systems. Next, we discuss the local well-posedness of Cauchy problem to the resulting systems in Sobolev spaces $H^s({\mathbb R})$ with $s>3/2$, the approach is based on the new invariant properties, certain estimates for transport equations of the system. In addition, blow up and wave-breaking to the Cauchy problem of a system are studied.

Citation: Min Zhao, Changzheng Qu. The two-component Novikov-type systems with peaked solutions and $H^1$-conservation law. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2857-2883. doi: 10.3934/cpaa.2020245
##### References:
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Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132.  doi: 10.1016/j.physleta.2008.10.050. [8] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2. [9] A. Constantin and W. Strauss, Stability of peakons, Commun. Pure Appl. Math., 53 (2009), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.3.CO;2-C. [10] R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differ. Equ., 196 (2003), 429-444.  doi: 10.1016/S0022-0396(03)00096-2. [11] A. Degasperis, D. D. Holm and A. W. Hone, A new integral equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474.  doi: 10.1023/A:1021186408422. [12] A. S. Fokas, P. J. Olver and P. Rosenau, A plethora of integrable bi-Hamiltonian equations, Nonlinear Differ. Equ., 26 (1996), 93-101.  doi: 10.1007/978-1-4612-2434-1_5. [13] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Physica D, 95 (1996), 229-243.  doi: 10.1016/0167-2789(96)00048-6. [14] Y. Fu, Y. Liu and C. Z. Qu, Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons, Math. Ann., 348 (2010), 415-448.  doi: 10.1007/s00208-010-0483-9. [15] Y. Fu and C. Z. Qu, Well posedness and blow-up solution for a new coupled Camassa-Holm equations with peakons, J. Math. Phys., 50 (2009), 1-25.  doi: 10.1063/1.3064810. [16] B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981), 47-66.  doi: 10.1016/0167-2789(81)90004-X. [17] X. G. Geng and B. Xue, An extension of integrable peakon equations with cubic nonlinearity, Nonlinearity, 22 (2009), 1847-1856.  doi: 10.1088/0951-7715/22/8/004. [18] A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479.  doi: 10.1088/0951-7715/25/2/449. [19] A. Himonas and D. Mantzavinos, The initial value problem for a Novikov system, J. Math. Phys., 57 (2016), 1-22.  doi: 10.1063/1.4959774. [20] D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys. A, 43 (2010), 1-20.  doi: 10.1088/1751-8113/43/49/492001. [21] D. D. Holm and R. I. Ivanov, Two-component CH system: inverse scattering, peakons and geometry, Inverse Probl., 27 (2011), 1-24.  doi: 10.1088/0266-5611/27/4/045013. [22] D. D. Holm, L. Ó Náraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E, 79 (2009), 1-25.  doi: 10.1103/PhysRevE.79.016601. [23] A. N. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ., 6 (2009), 253-289.  doi: 10.4310/DPDE.2009.v6.n3.a3. [24] A. N. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41 (2008), 1-11.  doi: 10.1088/1751-8113/41/37/372002. [25] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224. [26] J. Kang, X. C. Liu, P. J. Olver and C. Z. Qu, Liouville correspondences between integrable hierarchies, SIGMA Symmetry Integrability Geom. Methods Appl., 13 (2017), 1-26.  doi: 10.3842/SIGMA.2017.035. [27] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations, Springer, Berlin, Heidelberg, (1975), 25–70. doi: https://doi.org/10.1007/BFb0067080. [28] C. E. Kenig, G. G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7. [29] S. Y. Lai, Global weak solutions to the Novikov equation, J. Funct. Anal., 265 (2013), 520-544.  doi: 10.1016/j.jfa.2013.05.022. [30] H. M. Li, Y. Q. Li and Y. Chen, Bi-Hamiltonian structure of multi-component Novikov equation, J. Nonlinear Math. Phys., 21 (2014), 509-520.  doi: 10.1080/14029251.2014.975522. [31] Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differ. Equ., 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683. [32] X. C. Liu, Y. Liu and C. Z. Qu, Stability of peakons for the Novikov equation, J. Math. Pures Appl., 101 (2014), 172-187.  doi: 10.1016/j.matpur.2013.05.007. [33] H. Lundmark and J. Szmigielski, Dynamics of interlacing peakons in the Geng-Xue equation, J. Integr. Sys., 2 (2017), 1-65.  doi: 10.1093/integr/xyw014. [34] H. Lundmark and J. Szmigielski, An inverse spectral problem related to the Geng-Xue two-component peakon equation, Mem. Amer. Math. Soc., 244 (2016), 1-87.  doi: 10.1090/memo/1155. [35] V. Novikov, Generalizations of the Camassa–Holm equation, J. Phys. A, 42 (2009), 1-11.  doi: 10.1088/1751-8113/42/34/342002. [36] P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900. [37] C. Z. Qu and Y. Fu, Cauchy problem and peakons of a two-component Novikov system, Sci. China Math., (2019), 32pp. doi: https://doi.org/10.1007/s11425-019-9557-6. [38] J. F. Song, C. Z. Qu and Z. J. Qiao, A new integrable two-component system with cubic nonlinearity, J. Math. Phys., 52 (2011), 1-9.  doi: 10.1063/1.3530865. [39] F. Tiğlay, The periodic Cauchy problem for Novikov's equation, Int. Math. Res. Notes, 2011 (2011), 4633-4648. [40] X. L. Wu and Z. Y. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Super. Pisa C1. Sci., 11 (2012), 707-727. [41] Z. P. Xin and P. Zhang, On the weak solutions to shallow water equations, Commun. Pure Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.3.CO;2-X.

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##### References:
 [1] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661. [2] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5. [3] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. [4] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586. [5] A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91.  doi: 10.1007/PL00004793. [6] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328. [7] A. Constantin and R.I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132.  doi: 10.1016/j.physleta.2008.10.050. [8] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2. [9] A. Constantin and W. Strauss, Stability of peakons, Commun. Pure Appl. Math., 53 (2009), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.3.CO;2-C. [10] R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differ. Equ., 196 (2003), 429-444.  doi: 10.1016/S0022-0396(03)00096-2. [11] A. Degasperis, D. D. Holm and A. W. Hone, A new integral equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474.  doi: 10.1023/A:1021186408422. [12] A. S. Fokas, P. J. Olver and P. Rosenau, A plethora of integrable bi-Hamiltonian equations, Nonlinear Differ. Equ., 26 (1996), 93-101.  doi: 10.1007/978-1-4612-2434-1_5. [13] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Physica D, 95 (1996), 229-243.  doi: 10.1016/0167-2789(96)00048-6. [14] Y. Fu, Y. Liu and C. Z. Qu, Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons, Math. Ann., 348 (2010), 415-448.  doi: 10.1007/s00208-010-0483-9. [15] Y. Fu and C. Z. Qu, Well posedness and blow-up solution for a new coupled Camassa-Holm equations with peakons, J. Math. Phys., 50 (2009), 1-25.  doi: 10.1063/1.3064810. [16] B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981), 47-66.  doi: 10.1016/0167-2789(81)90004-X. [17] X. G. Geng and B. Xue, An extension of integrable peakon equations with cubic nonlinearity, Nonlinearity, 22 (2009), 1847-1856.  doi: 10.1088/0951-7715/22/8/004. [18] A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479.  doi: 10.1088/0951-7715/25/2/449. [19] A. Himonas and D. Mantzavinos, The initial value problem for a Novikov system, J. Math. Phys., 57 (2016), 1-22.  doi: 10.1063/1.4959774. [20] D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys. A, 43 (2010), 1-20.  doi: 10.1088/1751-8113/43/49/492001. [21] D. D. Holm and R. I. Ivanov, Two-component CH system: inverse scattering, peakons and geometry, Inverse Probl., 27 (2011), 1-24.  doi: 10.1088/0266-5611/27/4/045013. [22] D. D. Holm, L. Ó Náraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E, 79 (2009), 1-25.  doi: 10.1103/PhysRevE.79.016601. [23] A. N. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ., 6 (2009), 253-289.  doi: 10.4310/DPDE.2009.v6.n3.a3. [24] A. N. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41 (2008), 1-11.  doi: 10.1088/1751-8113/41/37/372002. [25] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224. [26] J. Kang, X. C. Liu, P. J. Olver and C. Z. Qu, Liouville correspondences between integrable hierarchies, SIGMA Symmetry Integrability Geom. Methods Appl., 13 (2017), 1-26.  doi: 10.3842/SIGMA.2017.035. [27] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations, Springer, Berlin, Heidelberg, (1975), 25–70. doi: https://doi.org/10.1007/BFb0067080. [28] C. E. Kenig, G. G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7. [29] S. Y. Lai, Global weak solutions to the Novikov equation, J. Funct. Anal., 265 (2013), 520-544.  doi: 10.1016/j.jfa.2013.05.022. [30] H. M. Li, Y. Q. Li and Y. Chen, Bi-Hamiltonian structure of multi-component Novikov equation, J. Nonlinear Math. Phys., 21 (2014), 509-520.  doi: 10.1080/14029251.2014.975522. [31] Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differ. Equ., 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683. [32] X. C. Liu, Y. Liu and C. Z. Qu, Stability of peakons for the Novikov equation, J. Math. Pures Appl., 101 (2014), 172-187.  doi: 10.1016/j.matpur.2013.05.007. [33] H. Lundmark and J. Szmigielski, Dynamics of interlacing peakons in the Geng-Xue equation, J. Integr. Sys., 2 (2017), 1-65.  doi: 10.1093/integr/xyw014. [34] H. Lundmark and J. Szmigielski, An inverse spectral problem related to the Geng-Xue two-component peakon equation, Mem. Amer. Math. Soc., 244 (2016), 1-87.  doi: 10.1090/memo/1155. [35] V. Novikov, Generalizations of the Camassa–Holm equation, J. Phys. A, 42 (2009), 1-11.  doi: 10.1088/1751-8113/42/34/342002. [36] P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900. [37] C. Z. Qu and Y. Fu, Cauchy problem and peakons of a two-component Novikov system, Sci. China Math., (2019), 32pp. doi: https://doi.org/10.1007/s11425-019-9557-6. [38] J. F. Song, C. Z. Qu and Z. J. Qiao, A new integrable two-component system with cubic nonlinearity, J. Math. Phys., 52 (2011), 1-9.  doi: 10.1063/1.3530865. [39] F. Tiğlay, The periodic Cauchy problem for Novikov's equation, Int. Math. Res. Notes, 2011 (2011), 4633-4648. [40] X. L. Wu and Z. Y. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Super. Pisa C1. Sci., 11 (2012), 707-727. [41] Z. P. Xin and P. Zhang, On the weak solutions to shallow water equations, Commun. Pure Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.3.CO;2-X.
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